bounds on the locating roman domination number in trees

A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the value f(V (G)) = ∑u∈V(G) f(u). An RDF f can be represented as f = (V0, V1, V2), where Vi = {v ∈ V : f(v) = i} for i = 0, 1, 2. An RDF f = (V0, V1, V2) is called a locating Roman dominating function (or just LRDF) if N(u) ∩ V2 ≠ N(v) ∩ V2 for any pair u, v of distinct vertices of V0. The locating Roman domination number γRL(G)$\gamma _R^L (G)$ is the minimum weight of an LRDF of G. In this paper, we study the locating Roman domination number in trees. We obtain lower and upper bounds for the locating Roman domination number of a tree in terms of its order and the number of leaves and support vertices, and characterize trees achieving equality for the bounds.